Tomography, or Computed Tomography (CT), is the process of forming a three-dimensional model of an object by combining 2D projections of the object of interest typically obtained through the use of any kind of penetrating particle or wave. Tomography is a rapidly advancing imaging technology with broad applications in such varied fields such as but not restricted to medicine, dentistry, biology, environmental, toxicology, mineralogy, and electronics. Tomographic processes use various tools, such as x-ray systems, transmission electron microscopes (TEM), scanning transmission electron microscopes (STEM), and/or atom probe microscopes (APM) to obtain various types of information such as, for example, atomic structure and chemical analysis of the sample. A 3D tomography dataset is typically obtained by back projecting a series of 2D images acquired through the sample at different angles, or in the case of an APM by reconstructing a volume from a sequence of field-evaporated atoms striking a position-sensitive detector.
Computed Tomography (CT) has been applied to the study of geological samples such as fossils, reservoir rocks and soils as a non-destructive image technique for the past three decades. The number of possible applications has grown with increasing resolution, with computed microtomography (μCT) imaging techniques now able reach sub-micron resolutions. One technique seeing increased application is multiple state imaging, in which the same sample is imaged under different conditions such as varying pressure, saturation states or weathering. When performing such analyses, a cylindrical container is required to maintain pressure or hold saturating fluid. For high-magnification, high con-angle imaging, such as that performed at the Australian National University (ANU) μCT facility by the present inventors, the sample is extremely close to the X-ray source. Containers are used in this case to dissipate the heat radiating from the source which to prevent sample movements. Containers are thus seeing increased use in μCT imaging. Along with such increased use, they introduce new challenges and opportunities, specifically with regard to the correction of what is referred to as “beam hardening” effects.
These deleterious effects are due to the scanned material's interaction with the scanning beam that may not be properly accounted for in the particular CT process employed for a particular scenario. The standard linearization pre-processing step that is performed prior to tomographic reconstruction is based on the Beer-Lambert law which assumes an exponential relationship between beam intensity and attenuation path length:I(t)=I0e−∫0tμ(x)dx,  (17)where I0 is the initial beam intensity, x is the depth into the material, μ(x) is the attenuation coefficient at a depth x and I(t) is the beam intensity after passing through a material thickness t. This assumes that the only effect of the material on the beam is to reduce its intensity; i.e. that the attenuation within each volume element is independent of how much material it has already passed through. However, the attenuation coefficient μ is also a function of x-ray beam energy, with attenuation generally decreasing with energy. Because of this, the lower energy components of a polychromatic beam (such as produced by lab-based microfocus x-resources), are preferentially absorbed, causing the beam to increase in average energy (become harder) as it passes through a sample. This effect is known as beam hardening and can cause artifacts such as cupping, (the apparent higher density near the edge of a sample), and streaking, (dark shadows between areas of relatively high-density). Such artifacts reduce image fidelity and can have a dramatic effect on downstream quantitative analysis, especially the process of segmentation in which regions of the tomogram are labeled to represent different materials.
There is, in general, insufficient information to perform tomographic reconstruction from polychromatic attenuation data, even when the beam spectrum and detector response is known exactly. Therefore, beam hardening presents a problem that seems computationally insoluble, unless assumptions can be made about the sample. While numerous methods have been proposed that target specific cases (such as imaging of bone and tissue), no method has seen widespread adoption for μCT imaging. Therefore, as other artifacts of x-ray imaging have been dramatically reduced indecent years, beam hardening is now the major source of imaging artifacts in many situations and it remains one of the key challenges for x-ray tomography. Provided herein are automated methods for beam hardening correction that work in situations that are of widespread interest for μCT imaging.
Various methods of correcting beam hardening artifacts in μCT images have been proposed. A common technique to reduce the artifacts is to filter the beam, where a sheet of material is placed before or after the sample to remove those low energy x-rays that would suffer rapid absorption within the sample. This method has a couple of shortcomings: First, it reduces severity of the artifacts but does not eliminate beam hardening. Further, it causes a reduction in overall flux reaching the detector, reducing signal to noise ratio and increasing required acquisition time for the same results. When a container is present, especially if it is highly attenuating as is required for high pressure imaging, the flux is reduced further, causing additional loss of contrast. A dual energy approach can also be taken, in which the same sample is imaged at two different energies, and the results combined under the assumption of a linear decomposition into basis functions. However this requires an additional μCT scan, which can be time consuming and expensive, especially if two scans are already required for dual-state imaging.
Other correction techniques occur after the acquisition of data. Beam hardening curve linearization corrects the projection data by assuming that the sample is composed of just one material. This is a reasonable assumption for samples which are approximately homogeneous, but cannot be used for heterogeneous materials or when a container is present. One suitable group of correction methods are the so-called post-reconstruction methods, which rely on reprojection of data to estimate the polychromatic and monochromatic projections. Reprojection uses forward-projection of rays to generate simulated projections from a reconstructed tomogram, based on knowledge about the x-ray spectrum and material properties. For monochromatic data containing no other artifacts, the reprojection and the original data will be the same. However, a reconstruction containing beam hardening artifacts is not, in general, fully consistent with the experimental data, so the reprojected data will be different from the experimental projections. Post-reconstruction methods use the difference between the experimental data and the simulated reprojection as a correction that can be applied to the data. However, even if the x-ray spectrum and detector response is known, one must have a perfect reconstruction of material attenuations. In addition, even for simple samples, the method fails for cylindrical samples where beam hardened projections will reconstruct into a completely consistent tomogram with commensurate beam hardening artifacts. One type of post-reconstruction beam hardening correction technique is proposed by Krumm et al (Krumm, M., Kasperl, S., and Franz, M., “Reducing non-linear artifacts of multi-material objects in industrial 3d computed tomography,” NDT & E International 41(4), 242-251 (2008)) which method will be referred to as Referenceless Post-Reconstruction Correction, or the RPC method. The RPC method allows the correction to operate with no knowledge of the materials or x-ray spectrum, but requires that the sample can be decomposed into homogeneous regions.